Properties

Label 4-96e4-1.1-c1e2-0-8
Degree $4$
Conductor $84934656$
Sign $1$
Analytic cond. $5415.50$
Root an. cond. $8.57846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·7-s − 12·17-s + 16·23-s − 8·25-s + 8·31-s − 4·41-s − 16·47-s + 34·49-s + 20·73-s + 24·79-s + 32·89-s + 16·97-s − 8·103-s − 32·113-s − 96·119-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 128·161-s + 163-s + 167-s − 8·169-s + ⋯
L(s)  = 1  + 3.02·7-s − 2.91·17-s + 3.33·23-s − 8/5·25-s + 1.43·31-s − 0.624·41-s − 2.33·47-s + 34/7·49-s + 2.34·73-s + 2.70·79-s + 3.39·89-s + 1.62·97-s − 0.788·103-s − 3.01·113-s − 8.80·119-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 10.0·161-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84934656\)    =    \(2^{20} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(5415.50\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 84934656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.233878867\)
\(L(\frac12)\) \(\approx\) \(5.233878867\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.891804870130875840848646473520, −7.83755570287223920399442511480, −7.22679572171406648156939886966, −6.74472797001150349431123940884, −6.51849290520016456220239335638, −6.51652409041043753850223347652, −5.72066994805876134102740261380, −5.18099467764377355964763643297, −4.99352759546942354614067172206, −4.85095349484092413470050416117, −4.58611259194452573823520787842, −4.26175440151506504749363871199, −3.66497928552278998574062224136, −3.32642231974352204512673566878, −2.59830341567481212214609881429, −2.31949074983506726454665680792, −1.83280724864054588998172517449, −1.72020307215401050376847078255, −0.997325790249093817029402849287, −0.56907382122675692397100036096, 0.56907382122675692397100036096, 0.997325790249093817029402849287, 1.72020307215401050376847078255, 1.83280724864054588998172517449, 2.31949074983506726454665680792, 2.59830341567481212214609881429, 3.32642231974352204512673566878, 3.66497928552278998574062224136, 4.26175440151506504749363871199, 4.58611259194452573823520787842, 4.85095349484092413470050416117, 4.99352759546942354614067172206, 5.18099467764377355964763643297, 5.72066994805876134102740261380, 6.51652409041043753850223347652, 6.51849290520016456220239335638, 6.74472797001150349431123940884, 7.22679572171406648156939886966, 7.83755570287223920399442511480, 7.891804870130875840848646473520

Graph of the $Z$-function along the critical line